We develop new structural results for apex-minor-free graphs and show their
power by developing two new approximation algorithms. The first is an
additive approximation for coloring within 2 of the optimal chromatic number,
which is essentially best possible, and generalizes the seminal result by
Thomassen [32] for bounded-genus graphs. This result also improves our
understanding from an algorithmic point of view of the venerable Hadwiger
conjecture about coloring H-minor-free graphs. The second
approximation result is a PTAS for unweighted TSP in apex-minor-free graphs,
which generalizes PTASs for TSP in planar graphs and bounded-genus graphs
[20, 2, 24, 15].

We strengthen the structural results from the seminal Graph Minor Theory of
Robertson and Seymour in the case of apex-minor-free graphs, showing that
apices can be made adjacent only to vortices if we generalize the notion of
vortices to “quasivortices” of bounded treewidth, proving a
conjecture from [10]. We show that this structure theorem is a powerful tool
for developing algorithms on apex-minor-free graphs, including for the classic
problems of coloring and TSP. In particular, we use this theorem to partition
the edges of a graph into k pieces, for any k, such that
contracting any piece results in a bounded-treewidth graph, generalizing
previous similar results for planar graphs [24] and bounded-genus graphs [15].
We also highlight the difficulties in extending our results to general
H-minor-free graphs.